3.53 \(\int \frac {x^3 (d+e x)}{(b x+c x^2)^2} \, dx\)

Optimal. Leaf size=45 \[ \frac {b (c d-b e)}{c^3 (b+c x)}+\frac {(c d-2 b e) \log (b+c x)}{c^3}+\frac {e x}{c^2} \]

[Out]

e*x/c^2+b*(-b*e+c*d)/c^3/(c*x+b)+(-2*b*e+c*d)*ln(c*x+b)/c^3

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Rubi [A]  time = 0.04, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \[ \frac {b (c d-b e)}{c^3 (b+c x)}+\frac {(c d-2 b e) \log (b+c x)}{c^3}+\frac {e x}{c^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(d + e*x))/(b*x + c*x^2)^2,x]

[Out]

(e*x)/c^2 + (b*(c*d - b*e))/(c^3*(b + c*x)) + ((c*d - 2*b*e)*Log[b + c*x])/c^3

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {x^3 (d+e x)}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {e}{c^2}+\frac {b (-c d+b e)}{c^2 (b+c x)^2}+\frac {c d-2 b e}{c^2 (b+c x)}\right ) \, dx\\ &=\frac {e x}{c^2}+\frac {b (c d-b e)}{c^3 (b+c x)}+\frac {(c d-2 b e) \log (b+c x)}{c^3}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 41, normalized size = 0.91 \[ \frac {\frac {b (c d-b e)}{b+c x}+(c d-2 b e) \log (b+c x)+c e x}{c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(d + e*x))/(b*x + c*x^2)^2,x]

[Out]

(c*e*x + (b*(c*d - b*e))/(b + c*x) + (c*d - 2*b*e)*Log[b + c*x])/c^3

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fricas [A]  time = 1.15, size = 69, normalized size = 1.53 \[ \frac {c^{2} e x^{2} + b c e x + b c d - b^{2} e + {\left (b c d - 2 \, b^{2} e + {\left (c^{2} d - 2 \, b c e\right )} x\right )} \log \left (c x + b\right )}{c^{4} x + b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

(c^2*e*x^2 + b*c*e*x + b*c*d - b^2*e + (b*c*d - 2*b^2*e + (c^2*d - 2*b*c*e)*x)*log(c*x + b))/(c^4*x + b*c^3)

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giac [A]  time = 0.17, size = 51, normalized size = 1.13 \[ \frac {x e}{c^{2}} + \frac {{\left (c d - 2 \, b e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{3}} + \frac {b c d - b^{2} e}{{\left (c x + b\right )} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

x*e/c^2 + (c*d - 2*b*e)*log(abs(c*x + b))/c^3 + (b*c*d - b^2*e)/((c*x + b)*c^3)

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maple [A]  time = 0.05, size = 61, normalized size = 1.36 \[ -\frac {b^{2} e}{\left (c x +b \right ) c^{3}}+\frac {b d}{\left (c x +b \right ) c^{2}}-\frac {2 b e \ln \left (c x +b \right )}{c^{3}}+\frac {d \ln \left (c x +b \right )}{c^{2}}+\frac {e x}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(e*x+d)/(c*x^2+b*x)^2,x)

[Out]

e*x/c^2-2/c^3*ln(c*x+b)*b*e+1/c^2*ln(c*x+b)*d-b^2/c^3/(c*x+b)*e+b/c^2/(c*x+b)*d

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maxima [A]  time = 0.83, size = 50, normalized size = 1.11 \[ \frac {b c d - b^{2} e}{c^{4} x + b c^{3}} + \frac {e x}{c^{2}} + \frac {{\left (c d - 2 \, b e\right )} \log \left (c x + b\right )}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)/(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

(b*c*d - b^2*e)/(c^4*x + b*c^3) + e*x/c^2 + (c*d - 2*b*e)*log(c*x + b)/c^3

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mupad [B]  time = 1.03, size = 56, normalized size = 1.24 \[ \frac {e\,x}{c^2}-\frac {b^2\,e-b\,c\,d}{c\,\left (x\,c^3+b\,c^2\right )}-\frac {\ln \left (b+c\,x\right )\,\left (2\,b\,e-c\,d\right )}{c^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(d + e*x))/(b*x + c*x^2)^2,x)

[Out]

(e*x)/c^2 - (b^2*e - b*c*d)/(c*(b*c^2 + c^3*x)) - (log(b + c*x)*(2*b*e - c*d))/c^3

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sympy [A]  time = 0.29, size = 44, normalized size = 0.98 \[ \frac {- b^{2} e + b c d}{b c^{3} + c^{4} x} + \frac {e x}{c^{2}} - \frac {\left (2 b e - c d\right ) \log {\left (b + c x \right )}}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x+d)/(c*x**2+b*x)**2,x)

[Out]

(-b**2*e + b*c*d)/(b*c**3 + c**4*x) + e*x/c**2 - (2*b*e - c*d)*log(b + c*x)/c**3

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